Theorem 2omotap 7257

Description: If there is at most one tight apartness on 2_o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)

Assertion

Ref	Expression
2omotap	⊢ (∃*𝑟 𝑟 TAp 2o → EXMID)

Proof of Theorem 2omotap

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2omotaplemst 7256	. . . . 5 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝑥 = {∅}) → 𝑥 = {∅})
2	1	ex 115	. . . 4 ⊢ (∃*𝑟 𝑟 TAp 2o → (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
3		df-stab 831	. . . 4 ⊢ (STAB 𝑥 = {∅} ↔ (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
4	2, 3	sylibr 134	. . 3 ⊢ (∃*𝑟 𝑟 TAp 2o → STAB 𝑥 = {∅})
5	4	adantr 276	. 2 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})
6	5	exmid1stab 4208	1 ⊢ (∃*𝑟 𝑟 TAp 2o → EXMID)

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 830   = wceq 1353  ∃*wmo 2027   ⊆ wss 3129  ∅c0 3422  {csn 3592  EXMIDwem 4194  2_oc2o 6410   TAp wtap 7247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-tr 4102  df-exmid 4195  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-1o 6416  df-2o 6417  df-pap 7246  df-tap 7248

This theorem is referenced by:  exmidmotap  7259